SummaryShifted linear systems are of the formwhere A ∈ C N ×N , b ∈ C N and {σ k } Nσ k=1 ∈ C is a sequence of numbers, called shifts. In order to solve (1) for multiple shifts efficiently, shifted Krylov methods make use of the shift-invariance property of their respective Krylov subspaces, i.e.and, therefore, compute one basis of the Krylov subspace (2) for all shifted systems. This leads to a significant speed-up of the numerical solution of the shifted problems because obtaining a basis of (2) is computational expensive. However, in practical applications, preconditioning of (1) is required, which in general destroys the shift-invariance. One of the few known preconditioners that lead to a new, preconditioned shifted problem is the so-called shift-and-invert preconditioner which is of the form (A − τ I) where τ is the seed shift. Most recently, multiple shift-and-invert preconditioners have been applied within a flexible GMRES iteration, cf. [10,20]. Since even the one-time application of a shift-and-invert preconditioner can be computationally costly, polynomial preconditioners have been developed for shifted problems in [1]. They have the advantage of preserving the shift-invariance (2) and being computational feasible at the same time.The presented work is a new approach to the iterative solution of (1). We use nested Krylov methods that use an inner Krylov method as a preconditioner for an outer Krylov iteration, cf. [17,24] for the unshifted case. In order to preserve the shift-invariance property, our algorithm only requires the inner Krylov method to produce collinear residuals of the shifted systems. The collinearity factor is then used in the (generalized) Hessenberg relation of the outer Krylov method. In this article, we present two possible combinations of nested Krylov algorithms, namely a combination of inner multi-shift FOM and outer multi-shift GMRES as well as inner multi-shift IDR(s) and outer multi-shift QMRIDR(s). However, we will point out that in principle every combination is possible as long as the inner Krylov method leads to collinear residuals. Since multi-shift IDR [31] does not lead to collinear residuals by default, the development of a collinear IDR variant that can be applied as an inner method within the nested framework is a second main contribution of this work.The new nested Krylov algorithm has been tested on several shifted problems. In particular, the inhomogeneous and time-harmonic linear elastic wave equation leads to shifts that directly correspond to different frequencies of the waves.
A new feature of glass-forming liquids, i.e., long-range density fluctuations of the order of 100 nm, has been extensively characterized by means of static light scattering, photon correlation spectroscopy and Rayleigh-Brillouin spectroscopy in orthoterphenyl (OTP) and 1,1-di(4(')-methoxy-5(')methyl-phenyl)-cyclohexane (BMMPC). These long-range density fluctuations result in the following unusual features observed in a light scattering experiment, which are not described by the existing theories: (i) strong q-dependent isotropic excess Rayleigh intensity, (ii) additional slow component in the polarized photon correlation function, and (iii) high Landau-Placzek ratio. These unusual features are equilibrium properties of the glass-forming liquids and depend only on temperature, provided that the sample has been equilibrated long enough. The temperature-dependent equilibration times were measured for BMMPC and are about 11 orders of magnitude longer than the alpha process. It was found that the glass-forming liquid OTP may occur in two states: with and without long-range density fluctuations ("clusters"). We have characterized the two states by static and dynamic light scattering in the temperature range from T(g) to T(g)+200 K. The relaxation times of the alpha process as well as the parameters of the Brillouin line are identical in both OTP with and without clusters. The alpha process (density fluctuations) in OTP was characterized by measuring either the polarized (VV) or depolarized (VH) correlation function, which are practically identical and q-independent. This feature, which is commonly observed in glass-forming liquids, is not fully explained by the existing theories.
In this work, we present a new numerical framework for the efficient solution of the time-harmonic elastic wave equation at multiple frequencies. We show that multiple frequencies (and multiple right-hand sides) can be incorporated when the discretized problem is written as a matrix equation. This matrix equation can be solved efficiently using the preconditioned IDR(s) method. We present an efficient and robust way to apply a single preconditioner using MSSS matrix computations. For 3D problems, we present a memory-efficient implementation that exploits the solution of a sequence of 2D problems. Realistic examples in two and three spatial dimensions demonstrate the performance of the new algorithm.
SummaryShifted linear systems are of the formwhere A ∈ C N ×N , b ∈ C N and {σ k } Nσ k=1 ∈ C is a sequence of numbers, called shifts. In order to solve (1) for multiple shifts efficiently, shifted Krylov methods make use of the shift-invariance property of their respective Krylov subspaces, i.e.and, therefore, compute one basis of the Krylov subspace (2) for all shifted systems. This leads to a significant speed-up of the numerical solution of the shifted problems because obtaining a basis of (2) is computational expensive. However, in practical applications, preconditioning of (1) is required, which in general destroys the shift-invariance. One of the few known preconditioners that lead to a new, preconditioned shifted problem is the so-called shift-and-invert preconditioner which is of the form (A − τ I) where τ is the seed shift. Most recently, multiple shift-and-invert preconditioners have been applied within a flexible GMRES iteration, cf. [10,20]. Since even the one-time application of a shift-and-invert preconditioner can be computationally costly, polynomial preconditioners have been developed for shifted problems in [1]. They have the advantage of preserving the shift-invariance (2) and being computational feasible at the same time.The presented work is a new approach to the iterative solution of (1). We use nested Krylov methods that use an inner Krylov method as a preconditioner for an outer Krylov iteration, cf. [17,24] for the unshifted case. In order to preserve the shift-invariance property, our algorithm only requires the inner Krylov method to produce collinear residuals of the shifted systems. The collinearity factor is then used in the (generalized) Hessenberg relation of the outer Krylov method. In this article, we present two possible combinations of nested Krylov algorithms, namely a combination of inner multi-shift FOM and outer multi-shift GMRES as well as inner multi-shift IDR(s) and outer multi-shift QMRIDR(s). However, we will point out that in principle every combination is possible as long as the inner Krylov method leads to collinear residuals. Since multi-shift IDR [31] does not lead to collinear residuals by default, the development of a collinear IDR variant that can be applied as an inner method within the nested framework is a second main contribution of this work.The new nested Krylov algorithm has been tested on several shifted problems. In particular, the inhomogeneous and time-harmonic linear elastic wave equation leads to shifts that directly correspond to different frequencies of the waves.
In the context of Galerkin discretizations of a partial differential equation (PDE), the modes of the classical method of Proper Orthogonal Decomposition (POD) can be interpreted as the ansatz and trial functions of a low-dimensional Galerkin scheme. If one also considers a Galerkin method for the time integration, one can similarly define a POD reduction of the temporal component. This has been described earlier but not expanded upon -probably because the reduced time discretization globalizes time which is computationally inefficient. However, in finite-time optimal control systems, time is a global variable and there is no disadvantage from using a POD reduced Galerkin scheme in time. In this paper, we provide a newly developed generalized theory for space-time Galerkin POD, prove its optimality in the relevant function spaces, show its application for the optimal control of nonlinear PDEs, and, by means of a numerical example with Burgers' equation, discuss the competitiveness by comparing to standard approaches.
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